copula (version 0.99-4)

tauAMH: Ali-Mikhail-Haq ("AMH")'s Kendall's Tau

Description

Compute Kendall's Tau of an Ali-Mikhail-Haq ("AMH") Archimedean copula with parameter theta. While that's analytically given explicitly, as $$1-\frac{2((1-\theta)^2\log(1-\theta) + \theta)}{3\theta^2},$$ for th$=\theta$; numerically, care has to be taken when $\theta \to 0$, avoiding accuracy loss already, for example, for $\theta$ as large as theta = 0.001.

Usage

tauAMH(th)

Arguments

th

numeric vector with values in $[0,1]$.

Value

• a vector of the same length as th ($= \theta$), with $\tau$ values in $[0, 1/3]$, of $\tau_A(\theta) = 1 - 2(\theta+(1-\theta)^2\log(1-\theta))/(3\theta^2)$, numerically accurately, to at least around 12 decimal digits.

Details

For small th ($=\theta$), we use Taylor series approximations of up to order 7, $$\tau_A(\theta) = \frac{2}{9}\theta(1 + \theta(\frac 1 4 + \frac{\theta}{10}(1 + \theta(\frac 1 2 + \theta \frac 2 7)))) + O(\theta^6),$$ where found that dropping the last two terms (e.g., only using 5 terms from the $k=7$ term Taylor polynomial) is actually numerically advantageous.

See Also

acopula-families, and their class definition, "acopula".

Examples

tauAMH(c(0, 2^-40, 2^-20))
curve(tauAMH, 0, 1)
curve(tauAMH, 1e-12, 1, log="xy") # linear, tau ~= 2/9*theta in the limit